Fractional Chern insulators (FCIs) are topologically ordered states of interacting fermions that share their universal properties with fractional quantum Hall states in Landau levels. FCIs have been found numerically in a variety of two-dimensional lattice models upon partially filling an almost dispersionless band with nontrivial topological character with repulsively interacting fermions. I will show how FCIs emerge in bands with Chern number C=1 and C=2 and in Z_2 topological insulators, where the latter are accompanied by a spontaneous breaking of time-reversal symmetry. Further, I will discuss the relevance of the noncommutative quantum geometry of the flat topological band to the stability of FCIs and how they can be distinguished from phases of spontaneously broken point group symmetry, such as charge density waves.